Quantum open system description of a hybrid plasmonic cavity

We present a unified quantum open system framework for lossy plasmonic cavities, treating coherent dynamics, relaxation, dephasing, and irreversible absorption on an equal footing. The Dyson equation for the cavity photon propagator in the random-phase approximation yields a complex self-energy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi mathvariant="script">S</a:mi> <a:mo>(</a:mo> <a:mi>ω</a:mi> <a:mo>)</a:mo> </a:mrow> </a:math> that accounts for both the renormalization and damping of hybrid plasmon-photon modes. It shows that increasing losses can drive a crossover from resolvable normal-mode splitting to a regime without resolved splitting, when the damping becomes comparable to or larger than the coherent hybridization scale. Tracing out the environment yields a Liouvillian for the upper polaritons (UPs) and lower polaritons (LPs) with leakage <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi mathvariant="normal">Γ</c:mi> <c:mo>=</c:mo> <c:mo>−</c:mo> <c:mn>2</c:mn> <c:mspace width="0.16em"/> <c:mi>Im</c:mi> <c:mspace width="0.16em"/> <c:mi mathvariant="script">S</c:mi> <c:mo>(</c:mo> <c:mi>ω</c:mi> <c:mo>)</c:mo> </c:mrow> </c:math> , internal UP <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mo>↔</h:mo> </h:math> LP scattering, and dephasing. Closed-form dynamics for populations and interbranch coherence provide analytic steady-state values, line shapes, and UP-LP quench rates, valid at low polariton density and in the ultrastrong-coupling regime. The theory is directly applicable to spectra, time-domain probes, and dissipation engineering in plasmonic and nanophotonic cavities.